0213_0231_1032_1230_1302_2031

Counting sequence:
1, 1, 2, 6, 18, 50, 135, 359, 944, 2464, 6402, 16588, 42917, 110971, 286946, 742298, 1921570, 4978510, 12910395, 33510783, 87061500, 226383708, 589136642, 1534297656, 3998484905, 10426612659, 27203438558, 71008408262, 185428263714, 484394646122, 1265783351951, 3308550132951, 8650040538856, 22619677709368, 59160262893058, 154752751327556, 404857230547213, 1059284394565803, 2771809912076794, 7253515319106194, 18982907094638370, 49682293303051942, 130035380029491651, 340360557386861039, 890906250916396660, 2332048420636285748, 6104553973118556674, 15980098983710370544, 41832395377175462097, 109509689316081881699, 286680327208220812342, 750495183613300543486, 1964725467671898305058, 5143505080645003264258, 13465400825103095839703, 35251838614484563603143, 92288219067433948873632, 241608633233224347807056, 632528442135674248733442, 1655956302215353551187004, 4335295461210106756240373, 11349830764260248536617691, 29713977662187569568717714, 77791618592005665799446538, 203659810959620368605104162, 533185459656132747675088510, 1395891372811192686154823627, 3654477196616379856606801247, 9567514928861721246104864620, 25048011799972003344611472268, 65576397392006174111935227138, 171680908855987087707206637096, 449465730198265493314936806777, 1176714960409234180992960877587, 3080676236259931385531966790926, 8065307318673267430926822244278, 21115231536674527365216799927586, 55280356005687755433658553261082, 144725767469921594963213403622559, 378896794181167932037380464271287, 991964279304322515217703416533720, 2596995303107255234054820860362472, 6799019996394329602954511474445314, 17800061082758015265989336316422580, 46601155304015272288006519469400157, 122003387298726874380409389216118795, 319408967925307926469931346831674794, 836223431190781909077338553880095746, 2189261137533975475083454253737684258, 5731559566498819627318639717048520406, 15005416646812733006983981685849596243, 39284688355449445015722372589131583439, 102848643967488665890596213554160957156, 269261233727463444947182594738299832420, 704935035556686549135630601101817381890, 1845543825172881086640194966647026675680, 4831696334600497491358471381114161665953, 12649544946242526068609790850586738708675, 33116937991575291440686247234309296601702, 86701267897998503445824593977544148412910, 226986863209024829881456260453823651681314

Generating function in Maple syntax:
(2*x^6-6*x^5+2*x^4-8*x^3+12*x^2-6*x+1)/(2*x-1)/(x^2-3*x+1)/(x^3+2*x-1)

Generating function in latex syntax:
\frac{2 x^{6}-6 x^{5}+2 x^{4}-8 x^{3}+12 x^{2}-6 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{3}+2 x -1\right)}

Generating function in sympy syntax:
(2*x**6 - 6*x**5 + 2*x**4 - 8*x**3 + 12*x**2 - 6*x + 1)/((2*x - 1)*(x**2 - 3*x + 1)*(x**3 + 2*x - 1))

Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x^3+2*x-1)*F(x)-2*x^6+6*x^5-2*x^4+8*x^3-12*x^2+6*x-1 = 0

Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{3}+2 x -1\right) F \! \left(x \right)-2 x^{6}+6 x^{5}-2 x^{4}+8 x^{3}-12 x^{2}+6 x -1 = 0

Explicit closed form in Maple syntax:
8957/295*Sum(_alpha^(-n+4),_alpha = RootOf(2*_Z^6-7*_Z^5+9*_Z^4-17*_Z^3+17*_Z^2-7*_Z+1))-54013/590*Sum(_alpha^(-n+3),_alpha = RootOf(2*_Z^6-7*_Z^5+9*_Z^4-17*_Z^3+17*_Z^2-7*_Z+1))+27198/295*Sum(_alpha^(-n+2),_alpha = RootOf(2*_Z^6-7*_Z^5+9*_Z^4-17*_Z^3+17*_Z^2-7*_Z+1))-12575/59*Sum(_alpha^(-n+1),_alpha = RootOf(2*_Z^6-7*_Z^5+9*_Z^4-17*_Z^3+17*_Z^2-7*_Z+1))+45572/295*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^6-7*_Z^5+9*_Z^4-17*_Z^3+17*_Z^2-7*_Z+1))-18413/590*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^6-7*_Z^5+9*_Z^4-17*_Z^3+17*_Z^2-7*_Z+1))+piecewise(n = 0,1,0)

Explicit closed form in latex syntax:
\frac{8957 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-17 \textit{\_Z}^{3}+17 \textit{\_Z}^{2}-7 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +4}\right)}{295}-\frac{54013 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-17 \textit{\_Z}^{3}+17 \textit{\_Z}^{2}-7 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{590}+\frac{27198 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-17 \textit{\_Z}^{3}+17 \textit{\_Z}^{2}-7 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{295}-\frac{12575 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-17 \textit{\_Z}^{3}+17 \textit{\_Z}^{2}-7 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{59}+\frac{45572 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-17 \textit{\_Z}^{3}+17 \textit{\_Z}^{2}-7 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{295}-\frac{18413 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-17 \textit{\_Z}^{3}+17 \textit{\_Z}^{2}-7 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{590}+\left(\left\{\begin{array}{cc}1 & n =0 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 50
a(6) = 135
a(n+6) = -2*a(n)+7*a(n+1)-9*a(n+2)+17*a(n+3)-17*a(n+4)+7*a(n+5), n >= 7

Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 18
a \! \left(5\right) = 50
a \! \left(6\right) = 135
a \! \left(n +6\right) = -2 a \! \left(n \right)+7 a \! \left(n +1\right)-9 a \! \left(n +2\right)+17 a \! \left(n +3\right)-17 a \! \left(n +4\right)+7 a \! \left(n +5\right), \quad n \geq 7

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/22527/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[16,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[16,x]*F[7,x]
F[7,x] = F[21,x]+F[8,x]
F[8,x] = F[20,x]+F[9,x]
F[9,x] = F[10,x]*F[17,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[13,x]*F[16,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = x
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[10,x]*F[16,x]*F[17,x]
F[20,x] = F[13,x]*F[2,x]
F[21,x] = F[22,x]+F[29,x]
F[22,x] = F[23,x]*F[25,x]
F[23,x] = F[1,x]+F[24,x]
F[24,x] = F[16,x]*F[23,x]
F[25,x] = F[26,x]
F[26,x] = F[16,x]*F[23,x]*F[27,x]
F[27,x] = F[1,x]+F[28,x]
F[28,x] = F[25,x]
F[29,x] = F[14,x]*F[2,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{16}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{21}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{20}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{17}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{10}\! \left(x \right) F_{16}\! \left(x \right) F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right) F_{2}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{29}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{25}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{16}\! \left(x \right) F_{23}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{16}\! \left(x \right) F_{23}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{25}\! \left(x \right)
F_{29}\! \left(x \right) = F_{14}\! \left(x \right) F_{2}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_16(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_16(x)*F_7(x))
Eq(F_7(x), F_21(x) + F_8(x))
Eq(F_8(x), F_20(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_17(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_13(x)*F_16(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x))
Eq(F_16(x), x)
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_10(x)*F_16(x)*F_17(x))
Eq(F_20(x), F_13(x)*F_2(x))
Eq(F_21(x), F_22(x) + F_29(x))
Eq(F_22(x), F_23(x)*F_25(x))
Eq(F_23(x), F_1(x) + F_24(x))
Eq(F_24(x), F_16(x)*F_23(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_16(x)*F_23(x)*F_27(x))
Eq(F_27(x), F_1(x) + F_28(x))
Eq(F_28(x), F_25(x))
Eq(F_29(x), F_14(x)*F_2(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 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